From: Jeff Schenck <schenck@xxxx.poulenc.schenck.net>
Subject: Is pink noise stationary?
Date: 29 Jan 1999 00:00:00 GMT
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A few days ago, I carelessly made the remark that pink noise is
non-stationary.  A couple people disagreed.  I hadn't given it much
thought before, so I did a little research.  My source of information
is an article in the _IEEE Proceedings_ of May 1995 by N. J. Kasdin.
Other references are cited therein.

Pink and brown noises have power law spectra, i.e., their spectra have
the form S(f) = 1 / f^a, where a=1 for pink noise and a=2 for brown.
It turns out that, because of the rapid increase at low frequency for
a>=1 ("infrared catastrophe"), noises with such spectra are
non-stationary.  Noise processes corresponding to spectra where 0<a<1
are stationary.  Kasdin uses fractional calculus to derive an
(idealized) LTI filter that, when driven by white noise, would produce
noise with the desired power law spectrum.  He then uses this filter
to approximate the asymptotic autocorrelation functions for different
values of a.  For a=1 and t>>tau,

        R(t,tau) ~= (1/2pi) (log 4t - log |tau|),   where

        R(t,tau) def= E{ x(t+tau/2) x(t-tau/2) }.
 

Certainly, approximations to pink noise based on realizable filters
are stationary, but an ideal process with the above autocorrelation
function is not.

I hope this is useful or at least interesting.

Jeff
 

------------------------------
Jeff Schenck
schenck@xxxx.slip.net
 

After Pinochet's arrest, will Henry Kissinger dare to travel abroad?

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